Central to several objective approaches to Bayesian model selection is the use of training samples (subsets of the data), so as to allow utilization of improper objective priors. The most common prescription for choosing training samples is to choose them to be as small as possible, subject to yielding proper posteriors; these are called minimal training samples.
When data can vary widely in terms of either information content or impact on the improper priors, use of minimal training samples can be inadequate.
Important examples include certain cases of discrete data, the presence of censored observations, and certain situations involving linear models and explanatory variables. Such situations require more sophisticated methods of choosing training samples. A variety of such methods are developed in this paper, and successfully applied in challenging situations.
We consider the use of Bayesian information criteria for selection of the graph underlying an Ising model. In an Ising model, the full conditional distributions of each variable form logistic regression models, and variable selection techniques for regression allow one to identify the neighborhood of each node and, thus, the entire graph. We prove high-dimensional consistency results for this pseudo-likelihood approach to graph selection when using Bayesian information criteria for the variable selection problems in the logistic regressions. The results pertain to scenarios of sparsity and fol…
Bayesian model selection with improper priors is not well-defined because of the dependence of the marginal likelihood on the arbitrary scaling constants of the within-model prior densities. We show how this problem can be evaded by replacing marginal log-likelihood by a homogeneous proper scoring rule, which is insensitive to the scaling constants. Suitably applied, this will typically enable consistent selection of the true model.
Real-life statistical samples are often plagued by selection bias, which complicates drawing conclusions about the general population. When learning causal relationships between the variables is of interest, the sample may be assumed to be from a distribution in a causal Bayesian network (BN) model under selection. Understanding the constraints in the model under selection is the first step towards recovering causal structure in the original model. The conditional-independence (CI) constraints in a BN model under selection have been already characterised; there exist, however, additional, non-…
In objective Bayesian model selection, no single criterion has emerged as dominant in defining objective prior distributions. Indeed, many criteria have been separately proposed and utilized to propose differing prior choices. We first formalize the most general and compelling of the various criteria that have been suggested, together with a new criterion. We then illustrate the potential of these criteria in determining objective model selection priors by considering their application to the problem of variable selection in normal linear models. This results in a new model selection objective…
In this note we introduce linear regression with basis functions in order to apply Bayesian model selection. The goal is to incorporate Occam's razor as provided by Bayes analysis in order to automatically pick the model optimally able to explain the data without overfitting.
Linear models with a growing number of parameters have been widely used in modern statistics. One important problem about this kind of model is the variable selection issue. Bayesian approaches, which provide a stochastic search of informative variables, have gained popularity. In this paper, we will study the asymptotic properties related to Bayesian model selection when the model dimension $p$ is growing with the sample size $n$. We consider $p\le n$ and provide sufficient conditions under which: (1) with large probability, the posterior probability of the true model (from which samples are …
We consider the problem of choosing between parametric models for a discrete observable, taking a Bayesian approach in which the within-model prior distributions are allowed to be improper. In order to avoid the ambiguity in the marginal likelihood function in such a case, we apply a homogeneous scoring rule. For the particular case of distinguishing between Poisson and Negative Binomial models, we conduct simulations that indicate that, applied prequentially, the method will consistently select the true model.
Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation. Specifically, we study the usefulness of continuous bimodal priors to model hypervariance parameters, and the effect scaling has on the posterior mean through its relationship to penalization. Several model selection strategies, some f…
We are deeply appreciative of the initiative of the editor, Marina Vanucci, in commissioning a discussion of our paper, and extremely grateful to all the discussants for their insightful and thought-provoking comments. We respond to the discussions in alphabetical order [arXiv:1409.5291].
We consider a Bayesian approach to variable selection in the presence of high dimensional covariates based on a hierarchical model that places prior distributions on the regression coefficients as well as on the model space. We adopt the well-known spike and slab Gaussian priors with a distinct feature, that is, the prior variances depend on the sample size through which appropriate shrinkage can be achieved. We show the strong selection consistency of the proposed method in the sense that the posterior probability of the true model converges to one even when the number of covariates grows nea…
This paper revisits the procedures suggested by Dudewicz and Dalal (1975) and Rinott (1978) which are designed for selecting the population with the highest mean among independent Gaussian populations with unknown and possibly different variances. In a previous paper Jacobovic and Zuk (2017) made a conjecture that the relative asymptotic efficiency of these procedures equals to the ratio of two certain sequences. This work suggests a quasi-Bayesian modelling of the problem under which this conjecture is valid. In addition, this paper motivates an open question regarding the extreme value distr…
The Bayesian method is noted to produce spuriously high posterior probabilities for phylogenetic trees in analysis of large datasets, but the precise reasons for this over-confidence are unknown. In general, the performance of Bayesian selection of misspecified models is poorly understood, even though this is of great scientific interest since models are never true in real data analysis. Here we characterize the asymptotic behavior of Bayesian model selection and show that when the competing models are equally wrong, Bayesian model selection exhibits surprising and polarized behaviors in large…
Bayesian methods - either based on Bayes Factors or BIC - are now widely used for model selection. One property that might reasonably be demanded of any model selection method is that if a model ${M}_{1}$ is preferred to a model ${M}_{0}$, when these two models are expressed as members of one model class $\mathbb{M}$, this preference is preserved when they are embedded in a different class $\mathbb{M}'$. However, we illustrate in this paper that with the usual implementation of these common Bayesian procedures this property does not hold true even approximately. We therefore contend that to us…
Regular vine copulas can describe a wider array of dependency patterns than the multivariate Gaussian copula or the multivariate Student's t copula. This paper presents two contributions related to model selection of regular vine copulas. First, our pair copula family selection procedure extends existing Bayesian family selection methods by allowing pair families to be chosen from an arbitrary set of candidate families. Second, our method represents the first Bayesian model selection approach to include the regular vine density construction in its scope of inference. The merits of our approach…
This article is an invited discussion of the article by Gronau and Wagenmakers (2018) that can be found at https://dx.doi.org/10.1007/s42113-018-0011-7.
We present a Bayesian model selection approach to estimate the intrinsic dimensionality of a high-dimensional dataset. To this end, we introduce a novel formulation of the probabilisitic principal component analysis model based on a normal-gamma prior distribution. In this context, we exhibit a closed-form expression of the marginal likelihood which allows to infer an optimal number of components. We also propose a heuristic based on the expected shape of the marginal likelihood curve in order to choose the hyperparameters. In non-asymptotic frameworks, we show on simulated data that this exac…
We deal with Bayesian inference for Beta autoregressive processes. We restrict our attention to the class of conditionally linear processes. These processes are particularly suitable for forecasting purposes, but are difficult to estimate due to the constraints on the parameter space. We provide a full Bayesian approach to the estimation and include the parameter restrictions in the inference problem by a suitable specification of the prior distributions. Moreover in a Bayesian framework parameter estimation and model choice can be solved simultaneously. In particular we suggest a Markov-Chain…
Objective priors for sequential experiments are considered. Common priors, such as the Jeffreys prior and the reference prior, will typically depend on the stopping rule used for the sequential experiment. New expressions for reference priors are obtained in various contexts, and computational issues involving such priors are considered.
In this article, we investigate large sample properties of model selection procedures in a general Bayesian framework when a closed form expression of the marginal likelihood function is not available or a local asymptotic quadratic approximation of the log-likelihood function does not exist. Under appropriate identifiability assumptions on the true model, we provide sufficient conditions for a Bayesian model selection procedure to be consistent and obey the Occam's razor phenomenon, i.e., the probability of selecting the "smallest" model that contains the truth tends to one as the sample size…
It is common practice to use Laplace approximations to compute marginal likelihoods in Bayesian versions of generalised linear models (GLM). Marginal likelihoods combined with model priors are then used in different search algorithms to compute the posterior marginal probabilities of models and individual covariates. This allows performing Bayesian model selection and model averaging. For large sample sizes, even the Laplace approximation becomes computationally challenging because the optimisation routine involved needs to evaluate the likelihood on the full set of data in multiple iterations…
Bayesian Variable Selection in Generalized Linear Models
University of Trento（特伦托大学）；Fondazione Bruno Kessler (FBK)（布鲁诺·孔塞雷基金会）；BCAM — Basque Center for Applied Mathematics（巴斯克应用数学中心）；Ikerbasque — Basque Foundation for Science（巴斯克科学基金会）
We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a truncated sparsity prior for variable selection, we provide a set of conditions that guarantee both variable-selection consiste…
In this paper we extend existing Bayesian methods for variable selection in Gaussian process regression, to select both the regression terms and the active covariates in the spatial correlation structure. We then use the estimated posterior probabilities to choose between relatively few modes through cross-validation, and consequently improve prediction.
Selecting a subset of variables for linear models remains an active area of research. This paper reviews many of the recent contributions to the Bayesian model selection and shrinkage prior literature. A posterior variable selection summary is proposed, which distills a full posterior distribution over regression coefficients into a sequence of sparse linear predictors.
In this paper, we derive a Bayesian model order selection rule by using the exponentially embedded family method, termed Bayesian EEF. Unlike many other Bayesian model selection methods, the Bayesian EEF can use vague proper priors and improper noninformative priors to be objective in the elicitation of parameter priors. Moreover, the penalty term of the rule is shown to be the sum of half of the parameter dimension and the estimated mutual information between parameter and observed data. This helps to reveal the EEF mechanism in selecting model orders and may provide new insights into the ope…
This note is a discussion of the article "Bayesian model selection based on proper scoring rules" by A.P. Dawid and M. Musio, to appear in Bayesian Analysis. While appreciating the concepts behind the use of proper scoring rules, including the inclusion of improper priors, we point out here some possible practical difficulties with the advocated approach.
Bayesian optimization is a popular tool for data-efficient optimization of expensive objective functions. In real-life applications like engineering design, the designer often wants to take multiple objectives as well as input uncertainty into account to find a set of robust solutions. While this is an active topic in single-objective Bayesian optimization, it is less investigated in the multi-objective case. We introduce a novel Bayesian optimization framework to efficiently perform multi-objective optimization considering input uncertainty. We propose a robust Gaussian Process model to infer…
This paper introduces the parallel hierarchical sampler (PHS), a Markov chain Monte Carlo algorithm using several chains simultaneously. The connections between PHS and the parallel tempering (PT) algorithm are illustrated, convergence of PHS joint transition kernel is proved and and its practical advantages are emphasized. We illustrate the inferences obtained using PHS, parallel tempering and the Metropolis-Hastings algorithm for three Bayesian model selection problems, namely Gaussian clustering, the selection of covariates for a linear regression model and the selection of the structure of…
Heckman selection model is the most popular econometric model in analysis of data with sample selection. However, selection models with Normal errors cannot accommodate heavy tails in the error distribution. Recently, Marchenko and Genton proposed a selection-t model to perform frequentist' robust analysis of sample selection. Instead of using their maximum likelihood estimates, our paper develops new Bayesian procedures for the selection-t models with either continuous or binary outcomes. By exploiting the Normal mixture representation of the t distribution, we can use data augmentation to im…
Kobe Steel, Ltd.（神户制钢所）；The Institute of Statistical Mathematics（统计数理研究所）；RIKEN AIP（理化学研究所AIP）
Bayesian optimization is an effective method to efficiently optimize unknown objective functions with high evaluation costs. Traditional Bayesian optimization algorithms select one point per iteration for single objective function, whereas in recent years, Bayesian optimization for multi-objective optimization or multi-point search per iteration have been proposed. However, Bayesian optimization that can deal with them at the same time in non-heuristic way is not known at present. We propose a Bayesian optimization algorithm that can deal with multi-objective optimization and multi-point searc…
Decision-making often involves ranking and selection. For example, to assemble a team of political forecasters, we might begin by narrowing our choice set to the candidates we are confident rank among the top 10% in forecasting ability. Unfortunately, we do not know each candidate's true ability but observe a noisy estimate of it. This paper develops new Bayesian algorithms to rank and select candidates based on noisy estimates. Using simulations based on empirical data, we show that our algorithms often outperform frequentist ranking and selection algorithms. Our Bayesian ranking algorithms y…
We consider Bayesian logistic regression models with group-structured covariates. In high-dimensional settings, it is often assumed that only small portion of groups are significant, thus consistent group selection is of significant importance. While consistent frequentist group selection methods have been proposed, theoretical properties of Bayesian group selection methods for logistic regression models have not been investigated yet. In this paper, we consider a hierarchical group spike and slab prior for logistic regression models in high-dimensional settings. Under mild conditions, we esta…
We consider the bridge linear regression modeling, which can produce a sparse or non-sparse model. A crucial point in the model building process is the selection of adjusted parameters including a regularization parameter and a tuning parameter in bridge regression models. The choice of the adjusted parameters can be viewed as a model selection and evaluation problem. We propose a model selection criterion for evaluating bridge regression models in terms of Bayesian approach. This selection criterion enables us to select the adjusted parameters objectively. We investigate the effectiveness of …
Motivated by examples from genetic association studies, this paper considers the model selection problem in a general complex linear model system and in a Bayesian framework. We discuss formulating model selection problems and incorporating context-dependent {\it a priori} information through different levels of prior specifications. We also derive analytic Bayes factors and their approximations to facilitate model selection and discuss their theoretical and computational properties. We demonstrate our Bayesian approach based on an implemented Markov Chain Monte Carlo (MCMC) algorithm in simul…
We consider a context-dependent ranking and selection problem. The best design is not universal but depends on the contexts. Under a Bayesian framework, we develop a dynamic sampling scheme for context-dependent optimization (DSCO) to efficiently learn and select the best designs in all contexts. The proposed sampling scheme is proved to be consistent. Numerical experiments show that the proposed sampling scheme significantly improves the efficiency in context-dependent ranking and selection.
Bayesian Neural Networks (BNNs) have recently received increasing attention for their ability to provide well-calibrated posterior uncertainties. However, model selection---even choosing the number of nodes---remains an open question. In this work, we apply a horseshoe prior over node pre-activations of a Bayesian neural network, which effectively turns off nodes that do not help explain the data. We demonstrate that our prior prevents the BNN from under-fitting even when the number of nodes required is grossly over-estimated. Moreover, this model selection over the number of nodes doesn't com…
We propose a novel Bayesian model selection technique on linear mixed-effects models to compare multiple treatments with a control. A fully Bayesian approach is implemented to estimate the marginal inclusion probabilities that provide a direct measure of the difference between treatments and the control, along with the model-averaged posterior distributions. Default priors are proposed for model selection incorporating domain knowledge and a component-wise Gibbs sampler is developed for efficient posterior computation. We demonstrate the proposed method based on simulated data and an experimen…
The notion of expense in Bayesian optimisation generally refers to the uniformly expensive cost of function evaluations over the whole search space. However, in some scenarios, the cost of evaluation for black-box objective functions is non-uniform since different inputs from search space may incur different costs for function evaluations. We introduce a cost-aware multi-objective Bayesian optimisation with non-uniform evaluation cost over objective functions by defining cost-aware constraints over the search space. The cost-aware constraints are a sorted tuple of indexes that demonstrate the …
Student-$t$ processes have recently been proposed as an appealing alternative non-parameteric function prior. They feature enhanced flexibility and predictive variance. In this work the use of Student-$t$ processes are explored for multi-objective Bayesian optimization. In particular, an analytical expression for the hypervolume-based probability of improvement is developed for independent Student-$t$ process priors of the objectives. Its effectiveness is shown on a multi-objective optimization problem which is known to be difficult with traditional Gaussian processes.
The class of chain event graph models is a generalisation of the class of discrete Bayesian networks, retaining most of the structural advantages of the Bayesian network for model interrogation, propagation and learning, while more naturally encoding asymmetric state spaces and the order in which events happen. In this paper we demonstrate how with complete sampling, conjugate closed form model selection based on product Dirichlet priors is possible, and prove that suitable homogeneity assumptions characterise the product Dirichlet prior on this class of models. We demonstrate our techniques u…
We present a multi-objective Bayesian optimisation algorithm that allows the user to express preference-order constraints on the objectives of the type "objective A is more important than objective B". These preferences are defined based on the stability of the obtained solutions with respect to preferred objective functions. Rather than attempting to find a representative subset of the complete Pareto front, our algorithm selects those Pareto-optimal points that satisfy these constraints. We formulate a new acquisition function based on expected improvement in dominated hypervolume (EHI) to e…
Challenging research in various fields has driven a wide range of methodological advances in variable selection for regression models with high-dimensional predictors. In comparison, selection of nonlinear functions in models with additive predictors has been considered only more recently. Several competing suggestions have been developed at about the same time and often do not refer to each other. This article provides a state-of-the-art review on function selection, focusing on penalized likelihood and Bayesian concepts, relating various approaches to each other in a unified framework. In an…
In the era of big data, variable selection is a key technology for handling high-dimensional problems with a small sample size but a large number of covariables. Different variable selection methods were proposed for different models, such as linear model, logistic model and generalized linear model. However, fewer works focused on variable selection for single index models, especially, for single index logistic model, due to the difficulty arose from the unknown link function and the slow mixing rate of MCMC algorithm for traditional logistic model. In this paper, we proposed a Bayesian varia…
A general Bayesian framework for model selection on random network models regarding their features is considered. The goal is to develop a principle Bayesian model selection approach to compare different fittable, not necessarily nested, models for inference on those network realisations. The criterion for random network models regarding the comparison is formulated via Bayes factors and penalizing using the mostwidely used loss functions. Parametrizations are different in different spaces. To overcome this problem we incorporate and encode different aspects of complexities in terms of observa…
Approximate Bayesian computation (ABC) is a class of algorithmic methods in Bayesian inference using statistical summaries and computer simulations. ABC has become popular in evolutionary genetics and in other branches of biology. However model selection under ABC algorithms has been a subject of intense debate during the recent years. Here we propose novel approaches to model selection based on posterior predictive distributions and approximations of the deviance. We argue that this framework can settle some contradictions between the computation of model probabilities and posterior predictiv…
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique（里昂大学，里昂高等师范学院，克洛德·贝尔纳大学，法国国家科学研究中心，物理实验室）；IRIT/INP-ENSEEIHT, University of Toulouse（图卢兹大学 IRIT/INP-ENSEEIHT）
Piecewise constant denoising can be solved either by deterministic optimization approaches, based on the Potts model, or by stochastic Bayesian procedures. The former lead to low computational time but require the selection of a regularization parameter, whose value significantly impacts the achieved solution, and whose automated selection remains an involved and challenging problem. Conversely, fully Bayesian formalisms encapsulate the regularization parameter selection into hierarchical models, at the price of high computational costs. This contribution proposes an operational strategy that …
Indiana University Network Science Institute（印第安纳大学网络科学研究所）
A central problem in analyzing networks is partitioning them into modules or communities. One of the best tools for this is the stochastic block model, which clusters vertices into blocks with statistically homogeneous pattern of links. Despite its flexibility and popularity, there has been a lack of principled statistical model selection criteria for the stochastic block model. Here we propose a Bayesian framework for choosing the number of blocks as well as comparing it to the more elaborate degree- corrected block models, ultimately leading to a universal model selection framework capable o…
In this paper, we consider Bayesian variable selection problem of linear regression model with global-local shrinkage priors on the regression coefficients. We propose a variable selection procedure that select a variable if the ratio of the posterior mean to the ordinary least square estimate of the corresponding coefficient is greater than $1/2$. Under the assumption of orthogonal designs, we show that if the local parameters have polynomial-tailed priors, our proposed method enjoys the oracle property in the sense that it can achieve variable selection consistency and optimal estimation rat…
The Yule-Simon distribution is usually employed in the analysis of frequency data. As the Bayesian literature, so far, ignored this distribution, here we show the derivation of two objective priors for the parameter of the Yule-Simon distribution. In particular, we discuss the Jeffreys prior and a loss-based prior, which has recently appeared in the literature. We illustrate the performance of the derived priors through a simulation study and the analysis of real datasets.
In this paper we propose to make Bayesian inferences for the parameters of the Lomax distribution using non-informative priors, namely the Jeffreys prior and the reference prior. We assess Bayesian estimation through a Monte Carlo study with 500 simulated data sets. To evaluate the possible impact of prior specification on estimation, two criteria were considered: the bias and square root of the mean square error. The developed procedures are illustrated on a real data set.
It is becoming increasingly important for machine learning methods to make predictions that are interpretable as well as accurate. In many practical applications, it is of interest which features and feature interactions are relevant to the prediction task. We present a novel method, Selective Bayesian Forest Classifier, that strikes a balance between predictive power and interpretability by simultaneously performing classification, feature selection, feature interaction detection and visualization. It builds parsimonious yet flexible models using tree-structured Bayesian networks, and samples…
While the Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) are powerful tools for model selection in linear regression, they are built on different prior assumptions and thereby apply to different data generation scenarios. We show that in the finite-dimensional case their respective assumptions can be unified within an augmented model-plus-noise space and construct a prior in this space which inherits the beneficial properties of both AIC and BIC. This allows us to adapt the BIC to be robust against misspecified models where the signal to noise ratio is low.
